Giroud replaced Alexis Sanchez at half-time against Everton on Saturday and scored a last-minute equaliser as Arsenal recovered from 2-0 down to earn a point at Goodison Park.
The Premier League clash ended on a sour note for the Frenchman, however, as he damaged his ankle in charging down a clearance from Sylvain Distin.
It is feared that the forward has sustained a broken ankle, but Wenger was giving little away when he faced the media on Tuesday, other than to confirm Giroud is unavailable for Wednesday's UEFA Champions League play-off second leg at the Emirates Stadium, after a goalless first leg in Turkey last week.
"He will not play, he will see a specialist," Wenger said.
"It can be [a long-term injury] but we will see."
Lukas Podolski's future at the club has been thrown into doubt recently, with the Germany international yet to feature in any of Arsenal's three matches so far this season.
Juventus and Wolfsburg are both rumoured to be interested in the 29-year-old, but Wenger suggested the forward will not be moving on.
"Podolski is in the squad [to face Besiktas] and will stay with us," he said.
As well as Giroud, Wenger confirmed that both Mikel Arteta and Kieran Gibbs will miss Wednesday's clash.
"Arteta is a short-term injury but is out," he said. "Kieran Gibbs will be back after the internationals [next week]."
'Obviously Messi is the best player in history and being compared to him means that you're doing things right, but I try to be myself': Barcelona star responds to comparisons with Argentine legend
'Darwin Nunez would be gold for Arsenal': Liverpool legend claims Gunners should have signed Uruguayan striker in January
